Coriolis Effect in Optics: Unified Geometric Phase and Spin-Hall Effect

Abstract

We examine the spin-orbit coupling effects that appear when a wave carrying intrinsic angular momentum interacts with a medium. The Berry phase is shown to be a manifestation of the Coriolis effect in a non-inertial reference frame attached to the wave. In the most general case, when both the direction of propagation and the state of the wave are varied, the phase is given by a simple expression that unifies the spin redirection Berry phase and the Pancharatnam-Berry phase. The theory is supported by the experiment demonstrating the spin-orbit coupling of electromagnetic waves via a surface plasmon nano-structure. The measurements verify the unified geometric phase, demonstrated by the observed polarization-dependent shift (spin-Hall effect) of the waves.PACS numbers: 03.65. Vf, 41.20.Jb, Introduction.-Geometric phase is an inherent feature of the polarization optics that appears under evolution of electromagnetic waves in inhomogeneous and anisotropic media [1]. There are two types of geometric phases: (i) the spin redirection Rytov-Vladimirskii-Berry phase associated with the parallel transport of the wave field under SO(3) variations of the direction of propagation of the wave [2,3] and (ii) the Pancharatnam-Berry phase that occurs under SU(2) manipulations with the polarization state of light [4,5]. In the general case, when both the direction of propagation and polarization state of the wave are varied, the geometric phase becomes more intricate [6,7,8,9], and a unified geometrical description of the phase requires the tricky Majorana representation [8].

Spin–orbit coupling in surface plasmon scattering by nanostructures

“…In the electronic SHE, electrons with opposite spins undertake different trajectories in a current flow 1 . In optics, photons with opposite spins-left or right circularly polarized (LCP or RCP) light-may propagate in different directions upon interaction with the material environment if spin-orbital coupling is engineered using, for example, metasurfaces [2][3][4] , Berry phase 5 , refractive index gradient 6 or near-field interference 7 .…”

Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence

“…Because the Dk-induced shift increases linearly with the transmission distance z, we experimentally measure the shifts at different observation planes far from the metamaterial and plot them in Figure 3. We designed the metamaterial with a period of d520 mm; therefore, the rotational rate is V5p/20 rad mm 21 . According to Equation (7), the k-induced shift upon transmission can be calculated as Dx52s 6 0.03165z.…”

“…It can be estimated from Figure 4a that the actual conversion efficiency approaches this ideal value. We also implement several sets of experiments to evaluate the conversion efficiency, which is found to be approximately 96.3% for the metamaterial with V5p/20 rad mm 21 . For the metamaterial with V52p/20 rad mm 21 , the measured conversion efficiency is approximately 75.5% which deviates considerably from the theoretical prediction.…”

“…41 and 43. Two metamaterials with V56p/20 rad mm 21 were prepared. The dimension of each sample is 6 mm36 mm.…”

“…[14][15][16][17] The photonic SHE describes the mutual influence of the photon spin (polarization) and the trajectory (orbital angular momentum) of light-beam propagation, i.e., the spin-orbit interaction (SOI), which results in two types of geometric phases: the Rytov-VladimirskiiBerry (RVB) phase and the Pancharatnam-Berry (PB) phase. [18][19][20][21][22] The RVB phase is associated with the evolution of the propagation direction of light. When a light beam reflects/refracts at a planar interface between different media, a SOI occurs, and the corresponding momentum-dependent RVB phase leads to a spin-dependent realspace (coordinate) shift, i.e., the photonic SHE.…”